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arXiv:0803.3359 · Physical Review A

Adiabatic condition and quantum geometric potential

Jianda Wu, Meisheng Zhao, Jian-lan Chen, Yong-de Zhang

Published Jun 25, 2008• 33 citations
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Abstract

In this paper, we present a $\text{U}(1)$-invariant expansion theory of the adiabatic process. As its application, we propose and discuss different sufficient adiabatic approximation conditions. In these conditions, we find an interesting invariant quantity referred to as the quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.

DOI
10.1103/physreva.77.062114

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In this paper, we present a U(1)\text{U}(1)U(1)-invariant expansion theory of the adiabatic process. As its application, we propose and discuss different sufficient adiabatic approximation conditions. In these conditions, we find an interesting invariant quantity referred to as the quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
The quantum geometric potential is a gauge invariant carrying novel geometric features between any two energy levels or bands in quantum systems. In generic time-dependent systems it gives a vital physical modification for the instantaneous energy gaps, laying down more appropriate quantum adiabatic conditions for both non-degenerate and degenerate systems. Remarkably, for generic parameterized quantum systems, the integration of the quantum geometric potential on a closed loop leads to a novel type of quantized winding number, which is a quantum counterpart of the Gauss-Bonnet theorem. The effects of the quantum geometric potential had been indirectly supported by the experiments on the quantum adiabatic evolution, however, a direct experimental observation so far is lacking. In this paper we propose an interference measurement to directly probe the quantum geometric potential, where the relevant parameters are easily accessible by current experimental apparatus. A direct confirmation of this new physical quantity could motive further theoretical and experimental investigations as well as its potential real applications.
An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.